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Haagerup's inequality for convolvers on free groups may be interpreted as aresult on $\tA_1$ buildings, i.e. trees. Here are proved analogous inequalitiesfor discrete groups acting freely on the vertices of $\tA_1\times\tA_1$ and$\tA_2$ buildings. The results apply in particular to groups of type-rotatingautomorphisms acting simply transitively on the vertices of such buildings.These results provide the first examples of higher rank groups with property(RD).

A Haagerup Inequality for $\tA_1\times\tA_1$ and $\tA_2$ Buildings